Maps preserving the \(\partial\)-spectrum of product or triple product of operators (Q6052770)

\par Let \({\mathcal B}(X)\) be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space \(X\). For any \(T\in{\mathcal B}(X)\), let \(\sigma(T)\) be its spectrum and \(\partial\sigma(T)\) be the boundary of its spectrum. A map \(\Delta\) from \({\mathcal B}(X)\) into the collection of all closed subsets of \(\mathbb{C}\) is said to be \(\partial\)-spectrum if \[\partial\sigma(T)\subseteq\Delta(T)\subseteq\sigma(T)\] for all \(T\in{\mathcal B}(X)\). In this paper under review, the authors characterize all surjective maps \(\phi\) on \({\mathcal B}(X)\) that satisfy \[\Delta(\phi(T)\phi(S))=\Delta(TS)\] for all \(S,~T\in{\mathcal B}(X)\). They also obtain a similar result but when the product \("TS"\) is replaced by the triple product \("TST"\).